Let S be a set and a binary operationMathworldPlanetmath on it. is said to be commutativePlanetmathPlanetmathPlanetmath if


for all a,bS.

Viewing as a functionMathworldPlanetmath from S×S to S, the commutativity of can be notated as


Some common examples of commutative operations are

  • addition over the integers: m+n=m+n for all integers m,n

  • multiplication over the integers: mn=mn for all integers m,n

  • addition over n×n matrices, A+B=B+A for all n×n matrices A,B, and

  • multiplication over the reals: rs=sr, for all real numbers r,s.

A binary operation that is not commutative is said to be non-commutative. A common example of a non-commutative operation is the subtraction over the integers (or more generally the real numbers). This means that, in general,


For instance, 2-1=1-1=1-2.

Other examples of non-commutative binary operations can be found in the attachment below.

Remark. The notion of commutativity can be generalized to n-ary operations, where n2. An n-ary operation f on a set A is said to be commutative if


for every permutation π on {1,2,,n}, and for every choice of n elements ai of A.

Title commutative
Canonical name Commutative
Date of creation 2013-03-22 12:22:45
Last modified on 2013-03-22 12:22:45
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 11
Author CWoo (3771)
Entry type Definition
Classification msc 20-00
Synonym commutativity
Synonym commutative law
Related topic Associative
Related topic AbelianGroup2
Related topic QuantumTopos
Related topic NonCommutativeStructureAndOperation
Related topic Subcommutative
Defines non-commutative